Approximately continuous function explained

In mathematics, particularly in mathematical analysis and measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the ordinary limit with an approximate limit.^[1] This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.^[2]

Definition

Let

be a Lebesgue measurable set, be a measurable function, and be a point where the Lebesgue density of is 1. The function is said to be approximately continuous at if and only if the approximate limit of at exists and equals .^[3]

Properties

A fundamental result in the theory of approximately continuous functions is derived from Lusin's theorem, which states that every measurable function is approximately continuous at almost every point of its domain.^[4] The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The Stepanov-Denjoy theorem provides a remarkable characterization:

Stepanov-Denjoy theorem: A function is measurable if and only if it is approximately continuous almost everywhere.^[5]

Approximately continuous functions are intimately connected to Lebesgue points. For a function

, a point is a Lebesgue point if it is a point of Lebesgue density 1 for and satisfies where denotes the Lebesgue measure and represents the ball of radius centered at . Every Lebesgue point of a function is necessarily a point of approximate continuity.^[6] The converse relationship holds under additional constraints: when is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.^[7]

See also

• Approximate limit
• Density point
• Density topology (which serves to describe approximately continuous functions in a different way, as continuous functions for a different topology)
• Lebesgue point
• Lusin's theorem
• Measurable function

Notes and References

1. Web site: Approximate continuity. Encyclopedia of Mathematics. January 7, 2025.
2. Book: Evans . L.C. . Gariepy . R.F. . Measure theory and fine properties of functions . CRC Press . Studies in Advanced Mathematics . Boca Raton, FL . 1992 .
3. Book: Federer, H. . Geometric measure theory . . Die Grundlehren der mathematischen Wissenschaften . 153 . New York . 1969 .
4. Book: Saks, S. . Theory of the integral . Hafner . 1952 .
5. 0528-2195. 103. 1. 95–96. Lukeš. Jaroslav. A topological proof of Denjoy-Stepanoff theorem. Časopis pro pěstování matematiky. 2025-01-20. 1978. 10.21136/CPM.1978.117963. free.
6. Book: Thomson, B.S. . Real functions . Springer . 1985 .
7. Book: Munroe, M.E. . Introduction to measure and integration . . 1953 .